Simplify and expand the following expression: $ \dfrac{4y}{y - 1}+\dfrac{y + 4}{5y + 1} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(y - 1)(5y + 1)$ Multiply the first term by $\dfrac{5y + 1}{5y + 1}$ $ \begin{align*} \dfrac{4y}{y - 1} \times \dfrac{5y + 1}{5y + 1} & = \dfrac{(4y)(5y + 1)}{(y - 1)(5y + 1)} \\ & = \dfrac{20y^2 + 4y}{(y - 1)(5y + 1)}\end{align*} $ Multiply the second term by $\dfrac{y - 1}{y - 1}$ $ \begin{align*} \dfrac{y + 4}{5y + 1} \times \dfrac{y - 1}{y - 1} & = \dfrac{(y + 4)(y - 1)}{(5y + 1)(y - 1)} \\ & = \dfrac{y^2 + 3y - 4}{(5y + 1)(y - 1)}\end{align*} $ Now we have: $ = \dfrac{20y^2 + 4y}{(y - 1)(5y + 1)} + \dfrac{y^2 + 3y - 4}{(5y + 1)(y - 1)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{20y^2 + 4y + y^2 + 3y - 4}{(y - 1)(5y + 1)} $ $ = \dfrac{21y^2 + 7y - 4}{(y - 1)(5y + 1)}$ Expand the denominator: $ = \dfrac{21y^2 + 7y - 4}{5y^2 - 4y - 1}$